hw6

  • Chris Nkinthorn, 20190707

Prompt: Consider the Couette flow between two parallel flat walls. The channel width is 2h. If the lower wall moves at a velocity U0 and the upper wall moves at a velocity U1.

Solution

Part a: Write down the governing equations of motion.

The governing equations are the conservation equations of mass, momentum, and energy. For a fluid, the velocity field is present in all three equations. This produces a total of five expressions as the scalar equations of continuity provide two, and the additional three compose the vector expression for momentum in each direction. However, as this is a two dimensional problem (2D), many of these expressions are zero.

From continuity, we see that the velocity vector, $\boxed{\vec{V} = u(y)\hat{i} }$.

Part b: Determine the velocity profile in the channel. Write the profile in a non-dimensional form.

Using the results of continuity, we then integrate conservation of momentum to produce two constants. Shifting from the centerline down to the lower wall, intuitively, $c_2$ is the lower wall velocity and $c_1$ is the difference between the two velocities. However, this must be expressed mathematically.

Part c: If the lower wall is at a temperature T0 and the upper wall is adiabatic, determine the temperature profile in non-dimensional form by including buoyant effects. This is another ODE which can be solved by direct integration

Part d: Determine the temperature distribution within the channel.

Part e: What is the heat flux at the lower wall?

Heat flux between a conducting solid and a convecting fluid is defined by the dimensionless Nusselt similarity parameter $Nu$, which is defined by evaluating the temperature gradient in the fluid at the surface.

Part f: What is the temperature of the upper wall?

Using the derived temperature distribution from part d, we evaluate $y =h$

Part g: Explain the meaning of any non-dimensional parameters occurring in the velocity or temperature profiles.

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